On-line Math 21

2.1  The definition of the derivative

Example 4 Find f¢(x) for the function f(x) = Öx .

Solution

f¢(x) = lim h® 0  f(x+h)-f(x) h = lim h® 0    ___ Öx+h   -Öx h = lim h® 0  æ è   ___ Öx+h   -Öx ö ø æ è   ___ Öx+h   +Öx ö ø h æ è   ___ Öx+h   +Öx ö ø = lim h® 0  (x+h)-x h(   ___ Öx+h   +Öx) = lim h® 0  1   ___ Öx+h   +Öx , cancelling out the h top and bottom, = 1 2Öx .

Finally, now that the limit is no longer a 0/0 form, you can take the limit as h® 0 .

Copyright (c) 2000 by David L. Johnson.